Nonseparating trees in 2-connected graphs and oriented trees in strongly connected digraphs

TL;DR

This paper verifies conjectures by Mader regarding the existence of specific subtrees and oriented trees in highly connected graphs and digraphs, extending known results to new classes of trees and connectivity conditions.

Contribution

The paper proves Mader's conjecture for two classes of trees in 2-connected graphs and for certain oriented stars and double-stars in strongly connected digraphs, under specified degree conditions.

Findings

Verified the conjecture for two classes of trees when k=2 in 2-connected graphs.

Proved the existence of oriented stars and double-stars in strongly connected digraphs with minimum semi-degree conditions.

Ensured the remaining graph after removal of these trees maintains strong connectivity.

Abstract

Mader [J. Graph Theory 65 (2010) 61-69] conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with $\delta(G)\geq \lfloor\frac{3}{2}k\rfloor+m-1$ contains a subtree $T'$ isomorphic to $T$ such that $G-V(T')$ is $k$-connected. The conjecture has been verified for paths, trees when $k=1$, and stars or double-stars when $k=2$. In this paper we verify the conjecture for two classes of trees when $k=2$. For digraphs, Mader [J. Graph Theory 69 (2012) 324-329] conjectured that every $k$-connected digraph $D$ with minimum semi-degree $\delta(D)=min\{\delta^+(D),\delta^-(D)\}\geq 2k+m-1$ for a positive integer $m$ has a dipath $P$ of order $m$ with $\kappa(D-V(P))\geq k$. The conjecture has only been verified for the dipath with $m=1$, and the dipath with $m=2$ and $k=1$. In this paper, we prove that every strongly connected digraph with minimum semi-degree $\delta(D)=min\{\delta^+(D),\delta^-(D)\}\geq m+1$ contains an oriented tree $T$ isomorphic to some given oriented stars or double-stars with order $m$ such that $D-V(T)$ is still strongly connected.